Geometric inequalities for electrostatic systems with boundary
Allan Freitas, Benedito Leandro, Ernani Ribeiro, Guilherme Sabo
TL;DR
The paper analyzes electrostatic systems with cosmological constant on compact manifolds with boundary, deriving sharp geometric inequalities that link boundary area, volume, and curvature to the interior electrostatic data. Utilizing the generalized Reilly formula and Pohozaev–Schoen identities, it proves rigidity results showing that equality in these bounds forces the de Sitter geometry, and it extends isoperimetric-type results to higher dimensions and to settings where the electric field is parallel to the lapse gradient. It also connects these geometric bounds to quasi-local masses, establishing Brown–York and charged Hawking mass inequalities with rigidity statements, thereby enriching the understanding of mass, boundary geometry, and charge in Einstein–Maxwell systems with $\Lambda$. The results are complemented by explicit higher-dimensional examples (e.g., RNdS, MP) and by a discussion of the role of boundary conditions (connected vs. disconnected, Einstein boundary) in attaining sharp bounds. Overall, the work advances higher-dimensional boundary geometry for electrostatic systems and clarifies when de Sitter space uniquely saturates the competing inequalities.
Abstract
In this article, we investigate electrostatic systems with a nonzero cosmological constant on compact manifolds with boundary. We establish new geometric properties for electrostatic manifolds in higher dimensions, extending previous results in the literature. Moreover, we prove sharp boundary estimates and isoperimetric-type inequalities for electrostatic manifolds, as well as volume and boundary inequalities involving the Brown-York and Hawking masses.